\(K\)-player additive extension of two-player games with an application to the Borda electoral competition game (Q1579043)

The paper introduces the notion of \(K\)-player additive extension of a symmetric two-player game, as follows: Let \(\{X,g\}\) be a symmetric two-player game. (This means that the strategy sets are \(S_1=S_2=X\), and \(g=G_1\) is the payoff function for player 1, with \(G_2(x,y)=G_1(y,x)\).) Given \(K\geq 2\), the \(K\)-player additive extension of \(\{X,g\}\) is the \(K\)-player normal form game \(\{(S_k,G_,)\}_{1\leq k\leq K}\) where for all \(k\) in \(\{1,\ldots, K\}\), \(S_k=X\), and \(G_k(x_1,\ldots,x_K)=\sum_{k'\neq k} g(x_k,x_k')\). If \(\{X,g\}\) is zero-sum, so is the \(K\)-player additive extension. The main theorem is that if \((x^*,x^*)\) is a unique mixed strategy equilibrium in the symmetric zero-sum game \(\{X,g\}\), then \((x^*,x^*,\ldots, x^*)\) is a unique equilibrium in the \(K\)-player additive extension. An example shows that the theorem does not extend to nonzero-sum games, and the importance of the symmetry assumption is discussed as well. The results are applied to a certain Borda electoral competition game.